Abstract
AbstractLet E be an elliptic curve defined over a number field F with good ordinary reduction at all primes above p, and let $F_\infty $ be a finitely ramified uniform pro-p extension of F containing the cyclotomic $\mathbb {Z}_p$ -extension $F_{\operatorname {cyc}}$ . Set $F^{(n)}$ be the nth layer of the tower, and $F^{(n)}_{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_p$ -extension of $F^{(n)}$ . We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda $ -invariant of the Selmer group over $F^{(n)}_{ \operatorname {cyc}}$ as $n\rightarrow \infty $ . This method has its origins in work of A. Cuoco, who studied $\mathbb {Z}_p^2$ -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.
Highlights
The Mordell–Weil Theorem states that given an elliptic curve E defined over a number field F, its F-rational points form a finitely generated abelian group, i.e., E(F) ≃ Zr ⊕ E(F)tors, where r is a non-negative integer called the Mordell–Weil rank
Given an abelian variety defined over a number field F with good ordinary reduction at the primes above p, Mazur showed that the rank of A is bounded in the cyclotomic Zp-extension of F
We study the growth of the rank of E(F(n)) by analyzing the growth of the λ-invariant of the Selmer group over Fc(ync) as n → ∞ using a generalizations of Kida’s formula due to Hachimori and Matsuno [11] and Lim [20]
Summary
The Mordell–Weil Theorem states that given an elliptic curve E defined over a number field F, its F-rational points form a finitely generated abelian group, i.e., E(F) ≃ Zr ⊕ E(F)tors, where r is a non-negative integer called the Mordell–Weil rank. For elliptic curves E/F , asymptotic formulas for the growth of the rank of E(F(n)) as n → ∞ have been proven by Lei and Sprung in [19] Such growth questions are studied in admissible uniform pro-p extensions of number fields by Delbourgo and Lei in [7], and by Hung and Lim in [14]. The method employed in this paper shows that in any context in which a satisfactory generalization of Kida’s formula is proved, it should be possible to analyze the growth of λ-invariants in noncommutative towers. We point out that analogs of Kida’s formula have been proven for fine Selmer groups by Kundu in [17] In this particular context, the number fields are assumed to be totally real. Such results were proved by Hatley and Lei in the supersingular setting, see [13]
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